View source for Liouville normal form

A way of writting a second-order ordinary linear differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596402.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596403.png" /> is parameter. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596404.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596406.png" />, then equation (1) reduces to the Liouville normal form (2) by means of the substitution

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596407.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596408.png" /></td> </tr></table>

which is called the Liouville transformation (introduced in [[#References|[1]]]). The Liouville normal form plays an important role in the investigation of the asymptotic behaviour of solutions of (1) for large values of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596409.png" /> or the argument, and in the investigation of the asymptotics of eigen functions and eigen values of the [[Sturm–Liouville problem|Sturm–Liouville problem]] (see [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Liouville,   ''J. Math. Pures Appl.'' , '''2'''  (1837)  pp. 16–35</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.C. Titchmarsh,   "Eigenfunction expansions associated with second-order differential equations" , '''1–2''' , Clarendon Press  (1946–1948)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR></table>